Question: Simplify; express your answer in exponential form. Assume $y\neq 0, q\neq 0$. $\dfrac{{(y^{-4}q^{2})^{-4}}}{{(y^{5}q^{-2})^{3}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(y^{-4}q^{2})^{-4} = (y^{-4})^{-4}(q^{2})^{-4}}$ On the left, we have ${y^{-4}}$ to the exponent ${-4}$ . Now ${-4 \times -4 = 16}$ , so ${(y^{-4})^{-4} = y^{16}}$ Apply the ideas above to simplify the equation. $\dfrac{{(y^{-4}q^{2})^{-4}}}{{(y^{5}q^{-2})^{3}}} = \dfrac{{y^{16}q^{-8}}}{{y^{15}q^{-6}}}$ Break up the equation by variable and simplify. $\dfrac{{y^{16}q^{-8}}}{{y^{15}q^{-6}}} = \dfrac{{y^{16}}}{{y^{15}}} \cdot \dfrac{{q^{-8}}}{{q^{-6}}} = y^{{16} - {15}} \cdot q^{{-8} - {(-6)}} = yq^{-2}$